Explanation Given: Fundamental Theorem of Calculus, R = [ a , b ] × [ c , d ] Proof: Step 1: By Fubini's theorem we get ∫ ∫ R f ( x , y ) d A = ∫ c d ∫ a b ∂ 2 F ∂ x ∂ y d x d y = ∫ c d ( ∫ a b ∂ ∂ x ( ∂ F ∂ y ) d x ) d y In the inner integral, y is considered as constant. Therefore, by the Fundamental Theorem of Calculus (part I) for the variable x, we have ∫ a b ∂ ∂ x ( ∂ F ∂ y ) d x = ∂ F ∂ y | x = b − ∂ F ∂ y | x = a = ∂ F ∂ y ( b , y ) − ∂ F ∂ y ( a , y ) Step 2: Using additivity of the single integral and using again the Fundamental Theorem of Calculus for the variable y and substituting them in integral obtained in Step 1, we get ∫ ∫ R f ( x , y ) d A = ∫ c d ∫ a b ∂ 2 F ∂ x ∂ y d x d y = x

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 16.1, Problem 52E

To determine

To prove:

The mathematical proof of the following extension of the Fundamental Theorem of Calculus to two variables:
If ∂2F∂x∂y=f(x,y) then ∫ ∫ R f(x,y)dA=F(b,d)−F(a,d)−F(b,c)+F(a,c)

where
R=[a,b]×[c,d]

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Chapter 16.1, Problem 51E

Chapter 16.1, Problem 53E

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Chapter 16 Solutions

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Knowledge Booster

To prove: The mathematical proof of the following extension of the Fundamental Theorem of Calculus to two variables: If ∂ 2 F ∂ x ∂ y = f ( x , y ) then ∫ ∫ R f ( x , y ) d A = F ( b , d ) − F ( a , d ) − F ( b , c ) + F ( a , c ) where R = [ a , b ] × [ c , d ]

Solution Summary: The author explains how Fubini's theorem extends to two variables.

To prove:

The mathematical proof of the following extension of the Fundamental Theorem of Calculus to two variables:
If ∂2F∂x∂y=f(x,y) then ∫ ∫ R f(x,y)dA=F(b,d)−F(a,d)−F(b,c)+F(a,c)

where
R=[a,b]×[c,d]

Want to see the full answer?

Chapter 16 Solutions

To prove: The mathematical proof of the following extension of the Fundamental Theorem of Calculus to two variables: If ∂ 2 F ∂ x ∂ y = f ( x , y ) then ∫ ∫ R f ( x , y ) d A = F ( b , d ) − F ( a , d ) − F ( b , c ) + F ( a , c ) where R = [ a , b ] × [ c , d ]

Solution Summary: The author explains how Fubini's theorem extends to two variables.

To prove: The mathematical proof of the following extension of the Fundamental Theorem of Calculus to two variables: If ∂2F∂x∂y=f(x,y) then ∫ ∫ R f(x,y)dA=F(b,d)−F(a,d)−F(b,c)+F(a,c) whereR=[a,b]×[c,d]

Want to see the full answer?

Chapter 16 Solutions

To prove:

The mathematical proof of the following extension of the Fundamental Theorem of Calculus to two variables:
If ∂2F∂x∂y=f(x,y) then ∫ ∫ R f(x,y)dA=F(b,d)−F(a,d)−F(b,c)+F(a,c)

where
R=[a,b]×[c,d]